In most transmission systems, in order to transmit at a high bit rate, symbols representing multiple bits are transmitted. The symbols are defined in the complex plane and are defined by a given constellation. The more bits that each symbol represents, the more points in the constellation. The complex symbols are transmitted with two frequency-domain signals, one representing the Real part and one representing the Imaginary part, known as the “in-phase” and “quadrature” signals.
FIG. 1, to which reference is now made, shows an exemplary 64-point constellation, for the QAM (quadrature amplitude modulation) method. Each point, which represents a symbol value, is a complex number having a Real and Imaginary parts. Thus, the point labeled 40 has a complex value of (5,3). It also has a complex bit pattern associated therewith, of 101 111, where 101 is the bit pattern associated with the Real part and 111 is the bit pattern associated with the Imaginary part of the symbol. Thus, there are 6 bits associated with each symbol.
FIG. 2, to which reference is now made, illustrates the transmission from one transmitting antenna, labeled 10, to another, receiving, antenna, labeled 12. Between the two there is a “channel” H, which is the path in the air between transmitting antenna 10 and receiving antenna 12, which may be obstructed, such as by trees, hills, buildings, etc. Transmitting antenna 10 transmits a symbol, here labeled u, and receiving antenna 12 receives a signal, here labeled y, which is a corrupted version of symbol u, by the channel H and by noise n, as follows:Y=hu+n  Equation 1
where the h defines how the channel affects transmission and n is random noise.
Equation 1 defines how the channel affects the transmission of a symbol. Point 5 in FIG. 1 shows a received symbol y. Note that point 5, while close to two points 7 and 9 of the constellation, doesn't sit on either of them. Thus, it is the job of the receiver to determine which symbol point 7 or 9 (or any other symbol) was originally transmitted.
A system where transmission of two or more symbols at the same time from two or more transmitter antennas to two or more receiver antennas takes place is known as a MIMO (multiple-input, multiple-output) communication system. The transmitted symbols are not related to each other and the paths, though similar, are not exactly the same. This is shown in FIG. 3, to which reference is now made, for the case of two transmitter antennas and two receiver antennas. Symbols u1 and u2 are transmitted from transmitting antennas 10a and 10b, respectively, and signals y1 and y2 are received at receiving antennas 12a and 12b, respectively. Both receiving antennas 12a and 12b receive versions of the transmitted symbols u1 and u2, which are transformed by the channel, where the channel for the MIMO case is a matrix H. and are affected by noise n1 and n2, respectively.
FIG. 4, to which reference is now made, shows exemplary received signals y1 and y2 in their constellations. Signal y1 is in the lower right quadrant while signal y2 is in the upper right quadrant. The goal is to decode signals y1 and y2 in order to determine transmitted symbols u1 and u2, which normally represent a coded version of original data bits including redundancy. The decoding process can be divided in two steps where, initially, the received signals y1 and y2 are converted to “soft” bit information such as Log Likelihood Ratios for each coded bit. Afterwards, the data bits are extracted from the soft bit information by a Maximum Likelihood Sequence Estimator, such as a soft-input Viterbi Decoder.